# Circle A has a radius of #4 # and a center of #(5 ,3 )#. Circle B has a radius of #5 # and a center of #(1 ,4 )#. If circle B is translated by #<2 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

##### 1 Answer

circles overlap.

#### Explanation:

What we have to do here is

#color(blue)"compare"# the distance (d) between the centres of the circles to the#color(blue)"sum of radii"# • If sum of radii > d, then circles overlap

• If sum of radii < d, then no overlap

Before calculating d we require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

#• " Under a translation "((2),(-1))#

#(1,4)to(1+2,4-1)to(3,3)larr" new centre of B"# To calculate d, use the

#color(blue)"distance formula"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))#

where# (x_1,y_1),(x_2,y_2)" are 2 coordinate points"# The 2 points here are (5 ,3) and (3 ,3)

let

# (x_1,y_1)=(5,3)" and " (x_2,y_2)=(3,3)#

#d=sqrt((3-5)^2+(3-3)^2)=sqrt4=2# sum of radii = radius of A + radius of B = 4 + 5 = 9

Since sum of radii > d, then circles overlap

graph{((x-5)^2+(y-3)^2-16)((x-3)^2+(y-3)^2-25)=0 [-20, 20, -10, 10]}